# Homotopic Jordan curves in C [pic included]

• mick25
In summary, the concept of a Jordan curve is that it's a simple closed curve that only intersects at the endpoints. It's the boundary of a set of two open sets: the "inside" (of the J. curve) is a bounded connected open set and the "outside" (of the J. curve) is an unbounded connected open set. Two closed curves, f1, f2 are homotopic in a connected open set A in C when we can continuously deform with closed curves fk; k in [0,1].
mick25
This isn't homework but I'm having trouble understanding the concept of non-homotopic and homotopic Jordan curves.

My understanding of Jordan curves and homotopy:

A Jordan curve is a simple closed curve (ie a closed curve that only intersects at the endpoints; f(z1)=f(z2) => z1=z2) such that its exterior is a disjoint union of two open sets:

i) the "inside" (of the J. curve) is a bounded connected open set
ii) the "outside" (of the J. curve) is an unbounded connected open set
iii) the said Jordan curve is the boundary of BOTH "inside" and "outside" (not true for general curves)

Two closed curves, f1, f2 are homotopic in a connected open set A in C when we can continuously deform with closed curves fk; k in [0,1].​

This is an example from my lecture notes:

Let A = C\<0>; ie. A is the complex plane with a "hole" at the origin

There are precisely 3 non-homotopic Jordan curves in A.

Here's my confusion:
thanks Office Shredder

Last edited:
The boundary of a set is the points you get in it's closure, if the origin is omitted from your topological space to begin with, it cannot possibly be in the closure of any set

Office_Shredder said:
The boundary of a set is the points you get in it's closure, if the origin is omitted from your topological space to begin with, it cannot possibly be in the closure of any set

Looks like I had a brain fart there haha, jeez I feel dumb. Thanks.

Also, I have another question if you could answer it: what makes a Jordan curve unique?

For example, there is this question: Sketch infinitely many non-homotopic Jordan curves in C\<-1, i, 1>.

I have these two curves f1 and f2: they have the same index at each point but aren't these two curves essentially the same?

They're not essentially the same (that is, they're not homotopic) because you can't continuously transform one into the other while staying in the set. That's what homotopic means. Remember that (-1,1,i) have been removed from the set. There's no way you can continuously morph the first curve into the second without having an intermediate curve that passes through i. That's not allowed since i is not in the set. Thus, they're not homotopic.

Think of the removed points as posts in the ground and the jordan curve as a loop of string. It's not a perfect analogy since technically you would need a string that could pass through itself, but it's still helpful for visualizing these problems. For the infinite number of non-homotopic curves, imagine dropping your loop of string around one post (-1), wrapping the loop and arbitrary number of times arouND -1 and i (in such a way that it doesn't cross itself), and then finally dropping the other end over the final post (+1).

Good luck on the midterm on Monday

## 1. What are homotopic Jordan curves?

Homotopic Jordan curves are simple closed curves in the complex plane C that are continuously deformable into each other without intersecting or passing through themselves.

## 2. How are homotopic Jordan curves different from ordinary Jordan curves?

Ordinary Jordan curves are simple closed curves that do not intersect or pass through themselves, but they may not necessarily be continuously deformable into each other. Homotopic Jordan curves, on the other hand, are not allowed to intersect or pass through themselves, and they must also be continuously deformable into each other.

## 3. What is the significance of homotopic Jordan curves in mathematics?

Homotopic Jordan curves are important in topology, the branch of mathematics that studies the properties of geometric objects that are preserved under continuous deformations. They are also used in complex analysis, a field of mathematics that deals with functions of complex numbers.

## 4. Can two homotopic Jordan curves have different shapes?

No, two homotopic Jordan curves must have the same shape and orientation. This means that they can only differ by a rotation, translation, or scaling of the entire curve.

## 5. How are homotopic Jordan curves related to the fundamental group of a space?

Homotopic Jordan curves are used to define the fundamental group of a space, which is a way of measuring the number of holes or voids in a given space. In particular, the fundamental group of a space can be calculated by considering all the possible homotopic Jordan curves in that space.

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